Abstract

The determination of the workspace is a very important
aspect from the design of a robot in order to specify its
margins of maneuver and to define all the points reachable
during its operation. In this article, we present the study of
the geometric reconfigurations of a constrained robot with
three degrees of freedom (D.O.F), which are obtained by
adjusting the length of the three kinematic chains and which
lead us to plot the working spaces of the robot. To do this,
we propose a method allowing to obtain a 3D representation
of the workspace. This method is exact in the sense that it
allows to determine the geometric nature of the boundary of
the workspace. We do not therefore use a discretization of
the translation parameters.

Keywords

Abstract

The determination of the workspace is a very important
aspect from the design of a robot in order to specify its
margins of maneuver and to define all the points reachable
during its operation. In this article, we present the study of
the geometric reconfigurations of a constrained robot with
three degrees of freedom (D.O.F), which are obtained by
adjusting the length of the three kinematic chains and which
lead us to plot the working spaces of the robot. To do this,
we propose a method allowing to obtain a 3D representation
of the workspace. This method is exact in the sense that it
allows to determine the geometric nature of the boundary of
the workspace. We do not therefore use a discretization of
the translation parameters.

Keywords

Introduction

Parallel architectures appeared about 50 years ago with
the Stewart-Gough rig and the first flight simulators. They
have since been used in other applications requiring more
precision, robustness, stiffness and the ability to handle heavy
objects with large accelerations or assembly in applications
that require high accuracy and control of contact efforts. .
The concept was recently adopted in the field of the machine
tool for high speed machining applications [1].

Reconfigurable robots are intelligent systems that can
autonomously change their configuration to accommodate
changing environments and tasks. The notion of
reconfiguration is used in areas as different as the automatic,
the electronics, the computer, the communications or the
manufacturing systems of production [2]. The main idea of
developing reconfigurable systems is based on the use of
modular components [3].

The interest in research and development in modular
robotics is guided by the need to find a balance between
cost and the ability to be multitasking, while improving the
effectiveness of action.

Selvakumar and Kumar [4] Offer a modular threesegment
parallel robot. Each segment contains four modules
where there are two actuator modules, one module which
constitutes a passive cylindrical seal and a module connected
to the end of the segment and containing a passive spherical joint. Mayaet al. [5] have taken over the well-known
Delta robot structure, which consists of three closed-loop
kinematic chains and proposed a new reconfigurable
structure by symmetrically adjusting the length of the
kinematic chains. The determination of the working space is
more complex than for the series manipulators, because of
the coupling between the translations and the orientations.

In this article, we propose a method to obtain a 3D
representation of the workspace for a given orientation.

Description of the Mechanical System

The parallel robot studied is a manipulator 3ddl
(degrees of freedom) which consists of a fixed base and a
mobile platform interconnected by three active segments
(motorized) and a passive central segment [6] (Figures 1
and 2).

Figure 1

prototype diagram.

Figure 1: prototype diagram.

Figure 2

Photo of four-segment constrained parallel robot.

Figure 2: Photo of four-segment constrained parallel robot.

The three active bars are connected to the movable base
by three d.d.l ball joint connections and to the fixed base
by two d.d.l universal joints; the three d.d.l passive central
bar may have several configurations or types of mechanical
structures, but we choose a PPP type structure to give the
manipulator a pure translation [7-10].

General Method of Obtaining the Reverse
Geometric Model

To determine the inverse geometrical model we consider
a reference X, Y, Z reinforcement, of center O, connected to
the fixed platform and to a reference frame U, V, W, of center
P, linked to the platform mobile form (Figure 1).

The active segments are linked to the fixed platform at
points A1, A2, A3 and connected to the mobile platform at
points B1, B2 and B3 Tel:

OAi−=ra ,PBi−=rb, i=1,2,3(1)

The coordinates of the vectors OAi and the PBi can be
written in the following form:

qa=f−1(X)(2)

OA−iA=aiA=[aixaiy0]T(3)

PB−iB=biB=[biubiv0]T(4)

aiA=[raCβiraSβi0]T,i=1,2,3(5)

biB=[rbCβirbSβi0]T,i=1,2,3(6)

The angle βi is measured between the X axis and the OAi
line and also between the U axis and the line PBi [8].

Où: Cβi and Sβi are respectively the cosine and the sine
of the angles βi.

We can write also:

biB=R(7)

Considering G the point of connection of the passive
segment to the fixed platform and H the point of connection
of the same segment to the mobile platform and Px, Py, Pz
the coordinates of the point P, used to define the position of
the mobile platform, we can write then:

OGA−=gA=[gxgy0]T(8)

PH−B=hB=[huhv0]T(9)

PA=OP−−=[PxPyPz]T(10)

Noting: ψ, θ and φ the angles RTL (Roulis-Tagage-Lacet)
defined by a succession of three rotations around the three
axes X, Y and Z, the matrix RBA
can thus be written as follows
[6]:

RBA=Rx(Φ)×Ry(θ)×Rz(ψ)(11)

To simplify the writing of the matrix of passage, we
replace in all that follows the cosines of the angles by C and
the sinuses by S, which makes it possible to write:

Equations (18) accurately represent the inverse
geometric model of our constrained robot. Therefore, given
the geometric dimensions of the parallel robot, we can
determine the lengths of the active segments that are needed
to move the mobile platform to the point of the working area
located at the coordinates (Px, Py, Pz).

Expressions (18) allow us to implement the robot
command based on the inverse geometric model. We pose:
qi=q2,

Workspace

The workspace of parallel robots is limited by three types
of constraints:

- The length of the active segments.

- Mechanical stresses on passive joints.

- Segment interference.

The general principle of calculating the workspace for
a constant-orientation manipulator is to assume that the
constraints on a chain i make it possible to define the volume
Vi that can be reached by the point Bi, the point of attachment
of the chain to the plateform. When the point Bi describes
this volume, the point P describes an identical volume Vip
obtained in translation Vi by the vector BiP which is constant
since the orientation is constant [11-14]. This volume is the
volume of work allowed for the point P with respect to the
constraints on the “i” segment. The workspace is the one
where the constraints on the whole segment are verified, it
is thus obtained by the intersection ViP. For simplicity, we
calculate the intersections of each of the Vip with the cutting
plane and we proceed to the intersection of the resulting
elements [6].

Each of the expressions (20) represents a sphere in the
space (x,y,z) of radius qi; the center of the mobile platform is
at the intersection of the three spheres.

The cylinders have a minimum length qimin and a maximum
length qimax, so the workspace is the set of intersections of
the three annular zones delimited by the boundaries of the
spheres of rays qimin and qimax.

The first equation delimits an annular zone whose
boundary is a central sphere (e62,0,0)
and radius q1 max and
the inner border a sphere of the same center and radius q1
min.

The second equation is a center sphere (e72,e42,0)
, it delimits
an annular zone whose outer radius boundary q2 max and
the inner boundary a sphere of the same center and radius
q2 max.

Similarly, if we consider the third equation, we see that it
delimits an annular zone whose outer boundary is a central
sphere. q3 max and radius q3 max , and the inner boundary, a
sphere of the same center and radius q3 max .

Reconfiguration

A geometrical reconfiguration of the Constraint
parallel robot reconfigurable to 3d.d.l is executed from the
inverse geometric model, adjusting simultaneously and
symmetrically the lengths of the three kinematic chains.

It is important to mention that each reconfiguration is
achieved by changing only the length of the respective link,
while maintaining the remaining lengths fixed [15-18].

Figures 4-6 show the resulting variations in the
working area for variations of the radius of the platform R,
corresponding to the values: 50, 250 and 350 mm.

Figure 4

Working area corresponding to R = 50 mm.

Figure 4: Working area corresponding to R = 50 mm.

Figure 5

orking area corresponding to R = 250 mm.

Figure 5: orking area corresponding to R = 250 mm.

Figure 6

Working area corresponding to R = 350 mm.

Figure 6: Working area corresponding to R = 350 mm.

It can be said that the maximum working area is obtained
when R=50 mm.

For the reconfiguration of the parameter q2 (or
parallelogram) from equations (19), a change of the working
space is obtained when the length of the link varies from
648 to 825, as can be seen, by increasing the value of q2, the
workspace increases by about the same proportion (Figure
7-9).

Although there is an increase in both the workspace and
the maximum range along the X, Y, and Z axes.

From equation (19), which express the articular variables
as a function of the position parameters of the platform. The
maximum displacement along the axis (X, Y, Z) is determined.

The design of a reconfiguration mechanism that
dynamically modifies the q2 parameter by means of a single
actuator seems rather difficult or almost impossible. On
the other hand, reconfiguring the R parameter causes a
significant change in the shape and volume of the workspace
(Figure 10-12).

Figure 10

Maximum displacement of the X axis (mm).

Figure 10: Maximum displacement of the X axis (mm).

Figure 11

Maximum displacements of the Y axis (mm).

Figure 11: Maximum displacements of the Y axis (mm).

Figure 12

Maximum displacement on the Z axis (mm).

Figure 12: Maximum displacement on the Z axis (mm).

Mechanism of reconfiguration

Our reconfiguration mechanism consists mainly of a
ball screw, rotated through a bevel gear pair and guided in translation by a sliding guide on a needle bearing (Figure
13).

Figure 13

Mechanism of reconfiguration.

Figure 13: Mechanism of reconfiguration.

Conclusion

The work presented in this article was devoted to the
inverse geometrical modeling of a reconfigurable constrained
parallel robot with 3 ddl and to the study of reconfigurations
as well as the variations of the working area. To accomplish
this task, we considered the reconfigurable parallel robot
constrained to 3d.d.l as a closed-loop structure. We chose
to model it in a different way by studying more closely the
structural characteristics of this robot. This study allowed us
to describe it in a systematic way, and to establish geometric
relations between the two elements that make up the
parallel structure: the platform and the segments (the active
segments and the passive segment).

However, we quickly calculated the reconfiguration of
the robot. To accomplish this task, we considered symmetry
of all kinematic chains, which allowed us to use a single
actuator.